Chapter 8, Problem 10Q

To determine

Check whether the given sequence is arithmetic, geometric, or neither. Find the rule for the $n\text{th}$ term of the sequence and the value of $a_9$ .

Answer to Problem 10Q

The $n\text{th}$ term of the sequence is $a_n=13+\left(n-1\right)\left(-7\right)$ and $a_9=-43$ .

Explanation of Solution

Given:

The given sequence is $13,6,-1,-8,---$

Calculation:

In an arithmetic sequence, the difference of consecutive terms, called the common difference, is constant.

Find the common difference between terms in the sequence $13,6,-1,-8,---$ .

  $\begin{array}{l}6-13=-7\\ -1-6=-7\\ -8-\left(-1\right)=-7\end{array}$

The common difference $\left(-7\right)$ is constant.

The general rule for the $n\text{th}$ term of the arithmetic sequence is $a_n=a_1+\left(n-1\right)d$ , where $a_1$ is first term and $d$ is the common difference.

Substitute the value of $a_1=13$ and $d=-7$ in the rule.

  $a_n=13+\left(n-1\right)\left(-7\right)$

Substitute the value of $n=9$ in the rule of $n\text{th}$ term.

  $\begin{array}{l}{a}_9=13+\left(9-1\right)\left(-7\right)\\ =13-56\\ =-43\end{array}$

Hence the $n\text{th}$ term of the sequence is $a_n=13+\left(n-1\right)\left(-7\right)$ and $a_9=-43$ .